This topic is not strictly linked to the numerical stability analysis but is more related to the conservation property in the discrete equation. Using care in the discretization of the convective terms allows for some property in the kinetic energy to be fulfilled. If the (induced) kinetic energy does not grow in time the scheme is more stable.

Have a look to this paper

Article Fully Conservative Higher Order Finite Difference Schemes fo...

Prof Denaro, thank you very much for your response and reference. From this reference, in conjunction with Pirozzoli's JCP 2010 paper, it so appears that certain split formulations with appropriate weighting coefficients could lead to schemes that conserves certain derived quantities at discrete level and hence the attained numerical stability.

Totally agree Tapan K. Sengupta

Title: Computational Design for Long-Term Numerical Integration of the Equations of Fluid Motion: Two-Dimensional Incompressible Flow. Part I

Authors:Arakawa, Akio

Affiliation: AA(University of California, Los Angeles, USA)

Publication:Journal of Computational Physics, Volume 1, Issue 1, p. 119-143. Publication Date: 08/1966

Origin: ADS; ELSEVIER DOI: 10.1016/0021-9991(66)90015-5

since that hystorical paper people have used the splitting with the goal to contrast kinetic energy pile-up at high wavenumbers. However, the paper focused on finite difference discretizations and the use of the vorticity variable for incompressible flows where the energy is not computed but induced by the solution of the momentum. Discrete conservation property using finite volume are a-priori guaranteed and there is no need at all to use the skew-symmetric form.

And the topic is further intriguing in the compressible flow assumption.

Dear Prof Sengupta,

Thank you for your valuable points and reference. Yes, I have your book and will go through it.

Dear Prof Denaro,

Thank you again for your response & full reference details .

As I understand it, skew-symmetric splitting enhances stability via both numerical and physical means. From the numerical/discrete standpoint, it -- along with other splittings -- has been shown to reduce aliasing errors (e.g., see Blaisdell's 1996 JCP paper or Kennedy/Grubber's 2008 JCP paper). Meanwhile from the physical standpoint, skew-symmetric splitting can be used to show implicit consistency with discrete forms of certain secondary equations (e.g., kinetic energy, enstrophy, etc...). In turn, adherence to secondary quantities can yield improved stability either directly (e.g., the kinetic energy for incompressible flows forms a bounded L2 estimate on the solution variable... see Morinishi's 1998 JCP paper) or indirectly in terms of more closely resembling the actual physical system (e.g., while kinetic energy is not bounded for compressible flows, a consistent representation may imply better accuracy of the entropy dynamics, which do form a bounded non-linear estimate on the solution system ... see Kuya et al.'s 2018 JCP paper).

Hi Ayaboe,

Thank you very much for taking time to explain points and quoting references. I'm currently referring these papers for a better understanding, and thank you for bringing to my attention Kuya et al.'s paper which I am yet to look at.

Best wishes.

Ananth, you are welcome! FYI, in case you havent come across it: there is also a Coppola et al. 2019 JCP paper that generalizes the different splittings relative to NS and their properties relative to primary and secondary conservation.

Sure Ayaboe, and I did go through it recently. Thank you :)

To tell the truth, my opinion is that the approach for compressible flows has different implication, this case has a direct solution of an energy equation, it is not implied by the solution of the momentum as is the case of incompressible flows.

Sure Prof Denaro, I understand.